The Conceptual Foundations of the Statistical Approach in Mechanics
By Paul Ehrenfest and Tatiana Ehrenfest
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About this ebook
Part One describes the older formulation of statistico-mechanical investigations (kineto-statistics of the molecule). Part Two takes up the modern formulation of kineto-statistics of the gas model, and Part Three explores W. B. Gibbs's major work, Elementary Principles in Statistical Mechanics and its coverage of such topics as the problem of axiomatization in kineto-statistics, the introduction of canonical and microcanonical distributions, and the analogy to the observable behavior of thermodynamic systems. The book concludes with the authors' original notes, a series of useful appendixes, and a helpful bibliography.
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The Conceptual Foundations of the Statistical Approach in Mechanics - Paul Ehrenfest
The Conceptual Foundations
of the Statistical Approach in
Mechanics
By Paul and Tatiana Ehrenfest
Translated by
Michael J. Moravcsik
DOVER PUBLICATIONS, INC.
Mineola, New York
Bibliographical Note
This Dover edition, first published by Dover Publications, Inc., in 1990 and brought back into print in 2015, is a reprint of the English translation by Michael J. Moravcsik published by The Cornell University Press, Ithaca, New York, in 1959.
International Standard Book Number
eISBN-13: 978-0-486-16314-7
Manufactured in the United States by Courier Corporation
66250001 2015
www.doverpublications.com
Foreword
IN recent years there has been a strong revival of interest in the foundations of statistical mechanics, and a great deal of important work has been done both in this country and abroad.
The Conceptual Foundations of Statistical Mechanics
is the title of a celebrated article which the late Paul Ehrenfest prepared in collaboration with his wife Tatiana Ehrenfest for the German Encyclopedia of Mathematical Sciences. The article appeared in 1912 and has since become a classic. In spite of the passage of time it has lost little of its scientific and didactic value, and no serious student of statistical mechanics can afford to remain ignorant of this great work.
Unfortunately it is not readily accessible and it requires a knowledge of German far beyond that of an average reader. It seemed therefore appropriate to make the article available in English, and the present translation, by Dr. Michael J. Moravcsik, is being offered to the public.
The translation adheres as rigidly as possible to the original. Because of this, the reader may be somewhat mystified by numbers like V 8 or VI (2) 21 which occasionally appear before references. These refer to other volumes of the Encyclopedia in which related articles have appeared. The bibliography at the end of the book is taken verbatim from the original, and no attempt has been made to bring it up to date. In this edition, the original notes are all collected at the end of the text. They are extremely important and contain a great deal of wisdom and wit, though the latter could not always be reflected in translation. A number of German words proved to be difficult to translate without sacrifice of nuances of meaning. They were left intact but italicized in the text and explained in footnotes.
It is hoped that the translation will be of help to student and teacher alike and that it will stimulate further work in this important and fascinating field.
M. KAC
G. E. UHLENBECK
June 1959
Authors’ Preface
THIS article is closely related to V 8 by L. Boltzmann and J. Nabl (Kinetische Theorie der Materie
). Both articles deal with the application of methods of probability theory in investigations of the motion of systems of molecules. However, whereas V 8 is concerned mainly with the physical results, the present work will take up the conceptual foundations of the procedure.
Since 1876 numerous papers have called attention to these foundations. In these papers the Boltzmann H- theorem, a central theorem of the kinetic theory of gases, was attacked. Without exception all studies so far published dealing with the connection of mechanics with probability theory grew out of the synthesis of these polemics and of Boltzmann’s replies. These discussions will therefore be referred to frequently in our report.
For the connections with V 3 by G. W. Bryan (Allgemeine Grundlagen der Thermodynamik
), with V 23 by W. Wien (Theorie der Strahlung
), and with VI(2) 21 by S. Oppenheim (Figur des Saturnringes
), see the last paragraphs of this article.
Preface to the Translation
THE great task of collecting the literature and of organizing the Encyklopädie article was done by Paul Ehrenfest. My contribution consisted only in discussing with him all the problems involved, and I feel that I succeeded in clarifying some concepts that were often incorrectly used. To this English translation of the Encyklopädie article, I would like to add the following remarks, which, in my opinion, should have been included in the original version of the article and for the omission of which I feel personally responsible.
1. At the time the article was written, most physicists were still under the spell of the derivation by Clausius of the second law of thermodynamics in the form of the existence of an integrating factor for the well-known expression for the quantity of heat ΔQ put into the system. In this derivation the irreversibility in time of all processes occurring in nature played an important role. Hence it seemed that the possibility of a reversal of the natural development (which according to the Wiederkehreinwand of Zermelo should occur after a sufficiently long time) threatened the validity of some of the most important results of thermodynamics. However, it became clear to me afterwards, that the existence of an integrating factor has to do only with the mathematical expression of ΔQ = dU+dA in terms of the differentials dx1, dx2, · · ·, dxn of the equilibrium parameters x1, x2, · · ·, xn and is completely independent of the direction in time of the development of the natural processes.¹ As a result, the fact of the reversibility of the mechanical motion, which is inescapable in the kinetic interpretation of the laws of thermodynamics, lost some of its importance. Nevertheless, even today many physicists are still following Clausius, and for them the second law of thermodynamics is still identical with the statement that the entropy can only increase.²
2. As a result of the above-mentioned point of view, the attention of most physicists was concentrated on the problem of the change of the H-function in time. Although Boltzmann had to admit that his Stosszahlansatz, on which the calculation of the change of H was founded, could not always be valid, he insisted that the probability of deviations was very small. This led to many efforts on the part of physicists to prove that one would always be more likely to find a system in the state of decreasing H than in the state of increasing H. The article of Paul and Tatiana Ehrenfest, "Über zwei bekannte Einwande gegen das Boltzmannsche H-Theorem" (Phys. Zeits., 1907), is in a way one of these attempts; it tries to show how one could reconcile the high probability of the validity of the Stosszahlansatz with the inescapable quasi-periodic recurrence of the same value of H.
3. It turned out that many physicists often confused two essentially different problems: (a) the relative probabilities of a decrease and of an increase of the H-function starting from a given value of H (which is different from the minimum value) and (b) the relative probabilities of a transition of the H-function from a higher to a lower value and of a transition in the opposite direction. The difference between these two problems I discussed in a paper, On a Misconception in the Probability Theory of Irreversible Processes
(Proc. Amst. Acad., vol. 38, 1925).
4. Since for the periods of increase the Stosszahlansatz cannot be valid, the question arises in which sense one should take the assertion of Boltzmann that deviations from the Stosszahlansatz are very improbable. As I see it now, the answer must be that for overwhelmingly long times when the H-function stays at its minimum value, neither of the objections against the H-theorems apply, and hence the Stosszahlansatz should be applicable, which explains the high probability of its occurrence. In this way the ingenious derivation of the formula for the change of H, that is, the H-theorem, can be rehabilitated, although by emphasizing another aspect than Boltzmann intended.
5. Although Boltzmann did not fully succeed in proving the tendency of the world to go to a final equilibrium state, there remain after all criticisms the following valuable results: first, the derivation of the Maxwell-Boltzmann distribution for equilibrium states, then the kinetic interpretation of the entropy by the H-function, and finally the explanation of the existence of an integrating factor for dU+dA. In thermodynamics the existence of such a factor is always based on an unexplained hypothesis.
The very important irreversibility of all observable processes can be fitted into the picture in the following way. The period of time in which we live happens to be a period in which the H-function of the part of the world accessible to observation decreases. This coincidence is really not an accident, since the existence and the functioning of our organisms, as they are now, would not be possible in any other period. To try to explain this coincidence by any kind of probability considerations will, in my opinion, necessarily fail. The expectation that the irreversible behavior will not stop suddenly is in harmony with the mechanical foundations of the kinetic theory.
T. EHRENFEST-AFANASSJEWA
Leiden, The Netherlands
February 1969
Contents
INTRODUCTION
1.Background
I. THE OLDER FORMULATION OF STATISTICO-MECHANICAL INVESTIGATIONS (KINETO-STATISTICS OF THE MOLECULE
2. The first provisional probability postulates
3. The equal frequency of apparently gleichberechtigt occurrences
3a. The assumptions of Clausius
3b. The Stosszahlansatz
4. The relative frequency of non-gleichberechtigt occurrences
4a. The qualitative assumptions and the first estimates of Clausius
4b. Maxwell’s derivation of a law for the distribution of velocities
4c. Boltzmann’s generalization of the Maxwell distribution law
5. Attempts to derive frequency postulates of the second kind from those of the first kind.
Appendix to Section 5
6. The Boltzmann H-theorem: The kinetic interpretation of irreversible processes
7. Objections to the result concerning irreversibility
7a. Loschmidt’s Umkehreinwand (1876)
7b. Zermelo’s Wiederkehreinwand
8. Closing remarks
II. THE MODERN FORMULATION OF STATISTICO-MECHANICAL INVESTIGATIONS (KINETO-STATISTICS OF THE GAS MODEL)
9. The mechanical properties of the gas model
9a. The gas model and its phase
9b. The phase space of the gas model (Γ- space)
9c. Liouville’s theorem
9d. Stationary density distributions in the Γ-space
10. The gas model as an ergodic system
10a. Ergodic mechanical systems
10b. Ergodic density distributions in Γ-space
11. The average behavior of the gas model for a motion of infinite duration
11a. Boltzmann’s investigations
11b. Criticism and meaning of Boltzmann’s results
12. Mechanical properties of the gas model (continued)
12a. The phase space of the molecule (µ- space): The state distribution Z of the molecule
12b. The volume in Γ-space corresponding to a state distribution Z
12c. Functions of the state distribution
12d. The function H(Z)
12e. The symbols dH(Z)/dt and AH(Z)/∆t
13. The dominance of the